9  Hazard function

The hazard function \(h\) describes the instantaneous risk of an event at a time \(t\). The instantaneuos risk at a time \(t\) is bounded between 0 and 1, where the sum over all \(t\) is 1. Mathematically, in the discrete case, this instantaneous risk equals the number of events between \(t\) and \(t+\Delta t\) divided by the population size at risk at time \(t\), divided by \(\Delta t\). In the continuous case, the hazard function is given as \[ \begin{aligned} h(t) &= \lambda(t) \\&= \lim_{\Delta t \to 0}\frac{\mathbb{P}\left(t \leq T < t + \Delta t \mid T \geq t\right)}{\Delta t} \end{aligned} \]

9.1 Cumulative hazard function

The cumulative hazard function \(H\) describes the cumulative hazard up until a time \(t\). \[ \begin{aligned} H(t) &= \int_0^t h(u)\,\textrm{d}u\\ &= \operatorname{ln}\left(S(t)\right) \end{aligned} \] As opposed to the survival function (see Chapter 8), the cumulative hazard function is monotonically increasing, i.e. \(H(t_2) \leq H(t_1)\) for all \(t_1 \geq t_2\).

9.2 Hazard ratio

The hazard ratio (HR) is often utilized when there is a desire to compare the hazard in one group to the hazard in another.

Non-collapsable. Not a causal estimand

  • \(\textrm{HR} < 1\): increased survival probability, compared to reference

  • \(\textrm{HR} = 1\): No difference in survival probability, compared to reference

  • \(\textrm{HR} > 1\): decreased survival probability, compared to reference

Remark 9.1. The confidence intervals for HRs are only symmetrical on the log-scale, and not the usually presented exp-scale.